Is the Unity API threadsafe?

I notice that Unity supports background loading of asset bundles, so I guess uses threads. Can I use threads in my own script and plugins? Are there any issues to be aware of?

No.

Every call you make into the Unity API, must be done from your main thread. It is possible for you to create your own threads, to make whatever calculations you like, however, the additional threads you create yourself should not call any Unity functions. They should communicate their results back to the mainthread, which can then call the Unity functions.

If you try to do this anyway, it might accidentally work, however future Unity updates (including those of the webplayer) are very likely to make this break.

You can achieve some of the advantages of threads by using coroutines - see the Coroutines and Yield overview in the Scripting Reference.